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The principal eigencurves for a nonselfadjoint elliptic operator

In this paper we study the existence of the principal eigencurves for a nonselfadjoint elliptic operator. We obtain their variational formulation. We establish also the continuity and the differentiability of the principal eigencurves. | Turk J Math 34 (2010) , 197 – 205. ¨ ITAK ˙ c TUB doi:10.3906/mat-0902-25 The principal eigencurves for a nonselfadjoint elliptic operator Aomar Anane, Omar Chakrone and Abdellah Zerouali Abstract In this paper we study the existence of the principal eigencurves for a nonselfadjoint elliptic operator. We obtain their variational formulation. We establish also the continuity and the differentiability of the principal eigencurves. Key Words: Nonsefadjoint elliptic operator , principl eigenvalue, principl eigencurve, Holland’s formula. 1. Introduction In this paper we consider the following problem ⎧ 1 ⎪ ⎪ ⎨ To find (λ, u) ∈ R × H (Ω) \ {0} such that (Pμ ) Lu − μm1 (x)u = λm2 (x)u ⎪ ⎪ ⎩ Bu = 0 in Ω, on ∂Ω, where Ω is a bounded C 1,1 domain in RN (N ≥ 1) with boundary ∂Ω, L is a second order elliptic operator of the form Lu := −div(A(x) +a0 (x)U, and B is a first order boundary operator of Neumann or Robin type: Bu := b(x), ∇u + b0 (x)u, where , denotes the scalar product in RN , the coefficient of L and B satisfy the condition where A(x) = (ai,j (x)) is a symmetric, uniformly positive definite N × N matrix, with ai,j ∈ C 0,1 (Ω), a and a0 ∈ L∞ (Ω), b and b0 ∈ C 0,1 (Ω), with b, ν > 0 (where ν is the unit exterior normal) and b0 ≥ 0 on ∂Ω, μ is a real parameter; and m1 and m2 ∈ L∞ (Ω) are possibly indefinite weights, with m1 and m2 ≡ 0 . The selfadjoint case (a ≡ 0 ) was considered by several authors, in particular P.A. Binding and Y. X. Huang in [1] , A. Dakkake and M. Hadda in [2] . For μ = 0 , the problem (Pμ ) was studied by T. Godoy, J. P. Gossez and 2000 AMS Mathematics Subject Classification: 35J20, 35J70, 35P05, 35P30. 197 ANANE, CHAKRONE, ZEROUALI S. Paczka in [3] . They gave a formula of minimax type (called Holland’s formula (cf., e.g., [6] )) for the principal eigenvalues of this problem. They gave also an application of this formula of minimax to the antimaximum principle. In this paper we study the existence of the principal eigencurves for .